I’m trying to understand how distributive lattices and coherent frames are equivalent. I think the functors have to look like this
- A coherent frame is send to the lattice $K(A)$ of finite elements and a coherent frame morphism $ A \rightarrow B$ is send to $K(A) \rightarrow K(B)$.
-A distributive lattice is send to the frame of ideals and a lattice morphism $f: L \rightarrow M$ is send to $f^{-1}: I(M)\rightarrow I(L)$, where $I(L)$ is the frame of ideals of $L$.
But the composition of the functors is contravariant, so there no natural isomorphism with the identity.
I believe a morphism of distributive lattices is sent to the direct image morphism, right adjoint to the inverse image. And the morphisms of coherent frames are restricted to preserve finite elements.